We consider rank 2 sheaves with \(\gamma =[2,-1,1-n)\) , discriminant \(\Delta =\frac{n}{2}-\frac{1}{8}\) .
The condition (REF ) gives \(\Delta \ge \delta _{\rm LP}(-\frac{1}{2})=\frac{5}{8}\)  for non-exceptional sheaves.
The generating function of Gieseker indices is given by [1]}
[2]}
\(\begin{aligned}h_{2,-1} &= q + (y^2+1+1/y^2)^2 q^2 + (y^8+2y^6+6y^4+9y^2+12+\dots ) q^3 \\& \quad + \left( y^{12}+2 y^{10}+6 y^8+13 y^6+24 y^4+35y^2+41+\dots \right) q^4+\dots \\&\rightarrow q + 9 q^2 + 48 q^3 +203 q^4 + 729 q^5 + 2346 q^6 + \dots \end{aligned}\) 
